RLT-POS: Reformulation-Linearization Technique (RLT)-based Optimization Software for Polynomial Programming Problems

Transcription

1 RLT-POS: Reformulation-Linearization Technique (RLT)-based Optimization Software for Polynomial Programming Problems E. Dalkiran 1 H.D. Sherali 2 1 The Department of Industrial and Systems Engineering, Wayne State University 2 Grado Department of Industrial and Systems Engineering, Virginia Tech MINLP 2014 Acknowledgement: This research has been supported by the National Science Foundation under Grant No. CMMI Dalkiran, Sherali (WSU, VT) RLT-based Optimization Software 1 / 25

2 Background Polynomial programming formulation Mathematical formulation: Problem PP PP: Minimize φ 0 (x) subject to where φ r (x) β r, r = 1,..., R 1 φ r (x) = β r, r = R 1 + 1,..., R Ax = b x Ω {x : 0 l j x j u j <, j N }, φ r (x) t T r α rt [ j J rt x j ], for r = 0,..., R. Dalkiran, Sherali (WSU, VT) RLT-based Optimization Software 2 / 25

3 Background Polynomial programming formulation Mathematical formulation: Problem PP PP: Minimize φ 0 (x) subject to where φ r (x) β r, r = 1,..., R 1 φ r (x) = β r, r = R 1 + 1,..., R Ax = b x Ω {x : 0 l j x j u j <, j N }, φ r (x) t T r α rt [ j J rt x j ], for r = 0,..., R. Reformulation Generate bound-factors and append bound-factor constraints: Bound-factors: (x j l j ) 0 and (u j x j ) 0, j N. Dalkiran, Sherali (WSU, VT) RLT-based Optimization Software 2 / 25

4 Background Mathematical formulation: Problem PP Polynomial programming formulation PP: Minimize φ 0 (x) where subject to φ r (x) β r, r = 1,..., R 1 φ r (x) = β r, r = R 1 + 1,..., R Ax = b x Ω {x : 0 l j x j u j <, j N }, φ r (x) t T r α rt [ j J rt x j ], for r = 0,..., R. Reformulation Generate bound-factors and append bound-factor constraints: Bound-factors: (x j l j ) 0 and (u j x j ) 0, j N. Bound-factor constraints: ( ) ( ) xj l j uj x j 0, (J1 J 2 ) N δ. j J 1 j J 2 Dalkiran, Sherali (WSU, VT) RLT-based Optimization Software 2 / 25

5 Background Mathematical formulation: Problem PP Polynomial programming formulation PP: Minimize φ 0 (x) where subject to φ r (x) β r, r = 1,..., R 1 φ r (x) = β r, r = R 1 + 1,..., R Ax = b x Ω {x : 0 l j x j u j <, j N }, φ r (x) t T r α rt [ j J rt x j ], for r = 0,..., R. Reformulation Generate bound-factors and append bound-factor constraints: Bound-factors: (x j l j ) 0 and (u j x j ) 0, j N. Bound-factor constraints: ( ) ( ) xj l j uj x j 0, (J1 J 2 ) N δ. j J 1 Linearization Substitute a new RLT variable for each distinct monomial as given by: j J 2 X J = j J x j, J δ N d. d=2 Dalkiran, Sherali (WSU, VT) RLT-based Optimization Software 2 / 25

6 Background Reformulation-Linearization Technique (RLT) RLT(Ω ): Minimize [φ 0 (x)] L subject to (2a) [φ r (x)] L β r, r = 1,..., R 1 (2b) [φ r (x)] L = β r, r = R 1 + 1,..., R (2c) Ax = b (2d) ( ) ) x j l j (u j x j 0, (J 1 J 2 ) N δ (2e) j J 1 j J 2 L x Ω {x : 0 l j x j u j <, j N } (2f) X J = j J x j, J δ d=2 N d. (2g) Dalkiran, Sherali (WSU, VT) RLT-based Optimization Software 3 / 25

7 Background Reformulation-Linearization Technique (RLT) RLT(Ω ): Minimize [φ 0 (x)] L subject to (2a) [φ r (x)] L β r, r = 1,..., R 1 (2b) [φ r (x)] L = β r, r = R 1 + 1,..., R (2c) Ax = b (2d) ( ) ) x j l j (u j x j 0, (J 1 J 2 ) N δ (2e) j J 1 j J 2 L x Ω {x : 0 l j x j u j <, j N } (2f) X J = j J x j, J δ d=2 N d. (2g) 1 Is there a strict subset of the fundamental bound-factor constraints for which the branch-and-bound algorithm described in Sherali and Tuncbilek [1992] would yet converge to a global optimum? 2 Is there additional valid inequalities that would tighten the RLT relaxation without sacrificing computational effort? Dalkiran, Sherali (WSU, VT) RLT-based Optimization Software 3 / 25

8 Background Reformulation-Linearization Technique (RLT) RLT(Ω ): Minimize [φ 0 (x)] L subject to (2a) [φ r (x)] L β r, r = 1,..., R 1 (2b) [φ r (x)] L = β r, r = R 1 + 1,..., R (2c) Ax = b (2d) ( ) ) x j l j (u j x j 0, (J 1 J 2 ) N δ (2e) j J 1 j J 2 L x Ω {x : 0 l j x j u j <, j N } (2f) X J = j J x j, J δ d=2 N d. (2g) 1 Is there a strict subset of the fundamental bound-factor constraints for which the branch-and-bound algorithm described in Sherali and Tuncbilek [1992] would yet converge to a global optimum? 2 Is there additional valid inequalities that would tighten the RLT relaxation without sacrificing computational effort? Model Enhancements: 1 The J-set of filtered bound-factor constraints, 2 Reduced RLT representations or RLT formulations in the reduced space, 3 ν-sdp cuts. Dalkiran, Sherali (WSU, VT) RLT-based Optimization Software 3 / 25

9 Model enhancements Constraint filtering routines The J-set of bound-factor constraints For a given index set J, N J J : the largest nonrepetitive set, d J : the cardinality of J. Dalkiran, Sherali (WSU, VT) RLT-based Optimization Software 4 / 25

10 Model enhancements Constraint filtering routines The J-set of bound-factor constraints For a given index set J, N J J : the largest nonrepetitive set, d J : the cardinality of J. Standard RLT constraints: j J 1 ( xj l j ) j J 2 ( ) uj x j L 0, (J 1 J 2 ) N J d J. Dalkiran, Sherali (WSU, VT) RLT-based Optimization Software 4 / 25

11 Model enhancements The J-set of bound-factor constraints Constraint filtering routines For a given index set J, N J J : the largest nonrepetitive set, d J : the cardinality of J. Standard RLT constraints: j J 1 ( xj l j ) j J 2 ( ) uj x j L 0, (J 1 J 2 ) N J d J. Proposition The convergence result in Sherali and Tuncbilek [1992] holds true if the following RLT bound-factor constraints are appended to the relaxation for each J such that the monomial j J x j appears in Problem PP: ( ) ( ) xj l j uj x j 0, (J 1 J 2 ) = J. j J 1 j J 2 L Dalkiran, Sherali (WSU, VT) RLT-based Optimization Software 4 / 25

12 Model enhancements The J-set and the standard RLT Constraint filtering routines The J-set of bound-factor constraints for x 2 1 x 2: [(x 1 l 1 )(x 1 l 1 )(x 2 l 2 )] L 0, [(x 1 l 1 )(u 1 x 1 )(x 2 l 2 )] L 0, [(u 1 x 1 )(u 1 x 1 )(x 2 l 2 )] L 0, [(x 1 l 1 )(x 1 l 1 )(u 2 x 2 )] L 0, [(x 1 l 1 )(u 1 x 1 )(u 2 x 2 )] L 0, [(u 1 x 1 )(u 1 x 1 )(u 2 x 2 )] L 0, Dalkiran, Sherali (WSU, VT) RLT-based Optimization Software 5 / 25

13 Model enhancements The J-set and the standard RLT Constraint filtering routines The J-set of bound-factor constraints for x 2 1 x 2: [(x 1 l 1 )(x 1 l 1 )(x 2 l 2 )] L 0, [(x 1 l 1 )(u 1 x 1 )(x 2 l 2 )] L 0, [(u 1 x 1 )(u 1 x 1 )(x 2 l 2 )] L 0, [(x 1 l 1 )(x 1 l 1 )(u 2 x 2 )] L 0, [(x 1 l 1 )(u 1 x 1 )(u 2 x 2 )] L 0, [(u 1 x 1 )(u 1 x 1 )(u 2 x 2 )] L 0, The bound-factor constraints for x 2 1 x 2 with the standard RLT: [(x 1 l 1 )(x 1 l 1 )(x 1 l 1 )] L 0, [(x 1 l 1 )(u 1 x 1 )(u 1 x 1 )] L 0, [(x 1 l 1 )(x 1 l 1 )(u 1 x 1 )] L 0, [(u 1 x 1 )(u 1 x 1 )(u 1 x 1 )] L 0. [(x 1 l 1 )(x 1 l 1 )(x 2 l 2 )] L 0, [(x 1 l 1 )(u 1 x 1 )(x 2 l 2 )] L 0, [(u 1 x 1 )(u 1 x 1 )(x 2 l 2 )] L 0, [(x 1 l 1 )(x 1 l 1 )(u 2 x 2 )] L 0, [(x 1 l 1 )(u 1 x 1 )(u 2 x 2 )] L 0, [(u 1 x 1 )(u 1 x 1 )(u 2 x 2 )] L 0, [(x 1 l 1 )(x 2 l 2 )(x 2 l 2 )] L 0, [(x 1 l 1 )(x 2 x 2 )(u 2 x 2 )] L 0, [(x 1 l 1 )(u 2 x 2 )(u 2 x 2 )] L 0, [(u 1 x 1 )(x 2 l 1 )(x 2 l 2 )] L 0, [(u 1 x 1 )(x 2 l 2 )(u 2 x 2 )] L 0, [(u 1 x 1 )(u 2 x 2 )(u 2 x 2 )] L 0, [(x 2 l 2 )(x 2 l 2 )(x 2 l 2 )] L 0, [(x 2 l 2 )(u 2 x 2 )(u 2 x 2 )] L 0, [(x 2 l 2 )(x 2 l 2 )(u 2 x 2 )] L 0, [(u 2 x 2 )(u 2 x 2 )(u 2 x 2 )] L 0. Dalkiran, Sherali (WSU, VT) RLT-based Optimization Software 5 / 25

14 Model enhancements Constraint filtering routines The number of RLT constraints and new RLT variables x j = j J x rj j j N J Dalkiran, Sherali (WSU, VT) RLT-based Optimization Software 6 / 25

15 Model enhancements Constraint filtering routines The number of RLT constraints and new RLT variables x j = j J x rj j j N J J-set N δ -set The number of bound-factor constraints j N (r j + 1) J The number of new RLT variables j N (r j + 1) ( N J + 1) J ( 2n + δ 1 ) ( n + δ ) δ δ (n + 1) Dalkiran, Sherali (WSU, VT) RLT-based Optimization Software 6 / 25

16 Model enhancements Constraint filtering routines The number of RLT constraints and new RLT variables x j = j J x rj j j N J J-set N δ -set The number of bound-factor constraints j N (r j + 1) J The number of new RLT variables j N (r j + 1) ( N J + 1) J ( 2n + δ 1 ) ( n + δ ) δ δ (n + 1) Example: For a PP involving only x1 3x 2x3 2 and x 4 2x 5x6 3 as nonlinear terms, the J-set and the N δ -set respectively generate in total: 48 and bound-factor constraints, 40 and 917 new RLT variables. Dalkiran, Sherali (WSU, VT) RLT-based Optimization Software 6 / 25

17 Model enhancements Reduced RLT representations Reduced RLT representations Linear Equality Subsystem: Ax = b Constraint-based RLT restrictions: [(Ax = b) j J x j ] L, yielding AX (. J) = bx J, J N d, d = 1,..., δ 1. (3) Dalkiran, Sherali (WSU, VT) RLT-based Optimization Software 7 / 25

18 Model enhancements Reduced RLT representations Reduced RLT representations Linear Equality Subsystem: Ax = b Constraint-based RLT restrictions: [(Ax = b) j J x j ] L, yielding AX (. J) = bx J, J N d, d = 1,..., δ 1. (3) Given a basis B of A, Ax = b Bx B + Nx N = b, AX (. J) = bx J, BX (BJ) + NX (NJ) = bx J, J N d, d = 1,..., δ 1. Dalkiran, Sherali (WSU, VT) RLT-based Optimization Software 7 / 25

19 Model enhancements Reduced RLT representations Reduced RLT representations Linear Equality Subsystem: Ax = b Constraint-based RLT restrictions: [(Ax = b) j J x j ] L, yielding AX (. J) = bx J, J N d, d = 1,..., δ 1. (3) Given a basis B of A, Ax = b Bx B + Nx N = b, AX (. J) = bx J, BX (BJ) + NX (NJ) = bx J, J N d, d = 1,..., δ 1. Proposition Let the equality system Ax = b be partitioned as Bx B + Nx N = b for any basis B of A, and define Z = (x, X) : Ax = b, (3) and X J = x j, J JN d, for d = 2,..., δ. j J Then, we have X J = j J x j, J N d, for d = 2,..., δ}. Dalkiran, Sherali (WSU, VT) RLT-based Optimization Software 7 / 25

20 Equivalent formulations Model enhancements Reduced RLT representations PP1: BX (BJ) + NX (NJ) = bx J, J N d, d = 1,..., δ 1 [ j J1 (x j l j ) j J2 (u j x j )] L 0, (J 1 J 2 ) N δ l x u X J = j J x j, J N d, d = 2,..., δ. Dalkiran, Sherali (WSU, VT) RLT-based Optimization Software 8 / 25

21 Equivalent formulations Model enhancements Reduced RLT representations PP1: BX (BJ) + NX (NJ) = bx J, J N d, d = 1,..., δ 1 [ j J1 (x j l j ) j J2 (u j x j )] L 0, (J 1 J 2 ) N δ l x u X J = j J x j, J N d, d = 2,..., δ. PP2: BX (BJ) + NX (NJ) = bx J, J N d, d = 1,..., δ 1 [ j J1 (x j l j ) j J2 (u j x j )] L 0, (J 1 J 2 ) J δ N l x u, and j J l j X J j J u j, J N d, d =2,...,δ, J B J 1 X J = j J x j, J JN d, d = 2,..., δ. Dalkiran, Sherali (WSU, VT) RLT-based Optimization Software 8 / 25

22 Equivalent formulations Model enhancements Reduced RLT representations PP1: BX (BJ) + NX (NJ) = bx J, J N d, d = 1,..., δ 1 [ j J1 (x j l j ) j J2 (u j x j )] L 0, (J 1 J 2 ) N δ l x u X J = j J x j, J N d, d = 2,..., δ. PP2: BX (BJ) + NX (NJ) = bx J, J N d, d = 1,..., δ 1 [ j J1 (x j l j ) j J2 (u j x j )] L 0, (J 1 J 2 ) J δ N l x u, and j J l j X J j J u j, J N d, d =2,...,δ, J B J 1 X J = j J x j, J JN d, d = 2,..., δ. Enhancing Algorithm RLT(PP2): Identify key bound-factor constraints, positive associated dual variables, at the root node and append within Problem PP2. Dalkiran, Sherali (WSU, VT) RLT-based Optimization Software 8 / 25

23 Model enhancements RLT(PP1) in R n m and Hybrid algorithm Reduced RLT representations RLT(PP1) in R n m : Eliminate the basic variables x B for the basis B via the substitution. Then, implement regular RLT process in the space of the (n m) nonbasic variables. Dalkiran, Sherali (WSU, VT) RLT-based Optimization Software 9 / 25

24 Model enhancements Reduced RLT representations RLT(PP1) in R n m and Hybrid algorithm RLT(PP1) in R n m : Eliminate the basic variables x B for the basis B via the substitution. Then, implement regular RLT process in the space of the (n m) nonbasic variables. RLT SDP (PP2) vs. RLT SDP (PP1) in R n m The size of the LP relaxations, The quality of the lower bounds at the root node. Dalkiran, Sherali (WSU, VT) RLT-based Optimization Software 9 / 25

25 Model enhancements Reduced RLT representations RLT(PP1) in R n m and Hybrid algorithm RLT(PP1) in R n m : Eliminate the basic variables x B for the basis B via the substitution. Then, implement regular RLT process in the space of the (n m) nonbasic variables. RLT SDP (PP2) vs. RLT SDP (PP1) in R n m The size of the LP relaxations, The quality of the lower bounds at the root node. RLT SDP (Hybrid) The swiftness of RLT SDP (PP1) in R n m, The robustness of RLT SDP (PP2). Compute µ = GAP 1 GAP 2 N 1 N 2 Dalkiran, Sherali (WSU, VT) RLT-based Optimization Software 9 / 25

26 Model enhancements Reduced RLT representations RLT(PP1) in R n m and Hybrid algorithm RLT(PP1) in R n m : Eliminate the basic variables x B for the basis B via the substitution. Then, implement regular RLT process in the space of the (n m) nonbasic variables. RLT SDP (PP2) vs. RLT SDP (PP1) in R n m The size of the LP relaxations, The quality of the lower bounds at the root node. RLT SDP (Hybrid) The swiftness of RLT SDP (PP1) in R n m, The robustness of RLT SDP (PP2). Compute µ = GAP 1 GAP 2 N 1 N 2 If (µ < 1) implement RLT SDP (PP2) else implement RLT SDP (PP1) in R n m. Dalkiran, Sherali (WSU, VT) RLT-based Optimization Software 9 / 25

27 Model enhancements Coordination between constraint filtering and reduced basis techniques PP: Minimize x 1 x 2 x 3 x5 2 subject to x x 3 + x 4 = 3 x 2 + x 5 = 6 x 3 x 5 x l i x i u i, i = 1, 2, 3, 4, 5. Dalkiran, Sherali (WSU, VT) RLT-based Optimization Software 10 / 25

28 Model enhancements Coordination between constraint filtering and reduced basis techniques PP: Minimize x 1 x 2 x 3 x5 2 subject to x x 3 + x 4 = 3 x 2 + x 5 = 6 x 3 x 5 x l i x i u i, i = 1, 2, 3, 4, 5. X X X 1355 = 0 X X X X 3555 = 0 X X X X 355 = 0. Dalkiran, Sherali (WSU, VT) RLT-based Optimization Software 10 / 25

29 Model enhancements Coordination between constraint filtering and reduced basis techniques PP: Minimize x 1 x 2 x 3 x5 2 subject to x x 3 + x 4 = 3 x 2 + x 5 = 6 x 3 x 5 x l i x i u i, i = 1, 2, 3, 4, 5. X X X 1355 = 0 X X X X 3555 = 0 X X X X 355 = 0. or X X X X 2355 = 0 X X X 3355 = 0 X X X 3455 = 0 X X X 355 = 0. Dalkiran, Sherali (WSU, VT) RLT-based Optimization Software 10 / 25

30 Model enhancements Coordination between constraint filtering and reduced basis techniques Reduced RLT Routine for Sparse Problems: Initialization: Define K {J : X J appears in (2a) (2c) and J J B 1}, K =, and L {J : X J appears within (2a) (2c) and J J B = }. Step 1: If K =, go to Step 4. Else, select an index set J K that involves the maximum number of basic variables, delete it from the list K and add it to the list K. Step 2: Let B j J be a randomly selected basic variable index. Multiply the representation of the basic variable x Bj in terms of the nonbasic variables by J {B j } x j, and linearize and append this to the relaxation. Step 3: If J {B j } involves any basic variable, the multiplication at Step 2 generates monomials involving basic variables. Include these monomials within the list K. Otherwise, if J {B j } does not involve any basic variable, then include the resulting monomials within the set L. Continue with Step 1. Step 4: Apply the J-set Routine to the set L in order to generate the proposed set of filtered bound-factor restrictions for reformulating the model. Dalkiran, Sherali (WSU, VT) RLT-based Optimization Software 11 / 25

31 Model enhancements Coordination between constraint filtering and reduced basis techniques PP: Minimize x 1 x 2 x 3 x 2 5 subject to x x 3 + x 4 = 3 x 2 + x 5 = 6 x 3 x 5 x l i x i u i, i = 1, 2, 3, 4, 5. J-RRLT: Minimize X subject to x x 3 + x 4 = 3 X X X X 3555 = 0 X X X X 355 = 0 x 2 + x 5 = 6 X X X 1355 = 0 X 35 X ( ) xj l j j J 1 j J 2 ( ) uj x j l i x i u i, i = 1, 2, 3, 4, 5, where L = {{4, 4}, {3, 3, 5, 5, 5}, {3, 4, 5, 5, 5}}. L 0, (J 1 J 2 ) L Dalkiran, Sherali (WSU, VT) RLT-based Optimization Software 12 / 25

32 Model enhancements Coordination between constraint filtering and reduced basis techniques J-set in R n m : Minimize 18X 355 3X X X X X subject to X 35 X l x 3 x 4 u 1 l 2 6 x 5 u 2 ( ) xj l j j J 1 j J 2 l i x i u i, i = 3, 4, 5. ( ) uj x j L 0, (J 1 J 2 ) L Table : Relaxation sizes and optimality gaps. # of equalities # of bound-factor constraints % optimality gap RLT RLT-E RRLT RRLT J-set J-RRLT J-NB Dalkiran, Sherali (WSU, VT) RLT-based Optimization Software 13 / 25

33 Model enhancements Coordination between constraint filtering and reduced basis techniques Table : The number of problems solved within the minimum CPU time among the J-set, J-RRLT+, and the J-NB. Degree J-set J-RRLT+ J-NB J-Hybrid (Offline) Total Dalkiran, Sherali (WSU, VT) RLT-based Optimization Software 14 / 25

34 Model enhancements Coordination between constraint filtering and reduced basis techniques Table : The number of problems solved within the minimum CPU time among the J-set, J-RRLT+, and the J-NB. Degree J-set J-RRLT+ J-NB J-Hybrid (Offline) Total Table : Average CPU time (in seconds) with the reduced basis techniques for sparse problems. Degree J-set J-RRLT+ J-NB J-Hybrid (Offline) Minimum J-Hybrid Dalkiran, Sherali (WSU, VT) RLT-based Optimization Software 14 / 25

35 ν-sdp cuts Model enhancements Semidefinite programming (SDP)-induced cuts (ν-sdp cuts) [xx T ] is symmetric and positive semidefinite M 0 = [xx T ] L 0. Dalkiran, Sherali (WSU, VT) RLT-based Optimization Software 15 / 25

36 ν-sdp cuts Model enhancements Semidefinite programming (SDP)-induced cuts (ν-sdp cuts) [xx T ] is symmetric and positive semidefinite M 0 = [xx T ] L 0. A stronger [ implication ] in this same vein is: [ 1 x (1) =, and defining the matrix M x 1 [x (1) x(1) T 1 x ] T L = x M 0 ] 0. Dalkiran, Sherali (WSU, VT) RLT-based Optimization Software 15 / 25

37 ν-sdp cuts Model enhancements Semidefinite programming (SDP)-induced cuts (ν-sdp cuts) [xx T ] is symmetric and positive semidefinite M 0 = [xx T ] L 0. A stronger [ implication ] in this same vein is: [ 1 x (1) =, and defining the matrix M x 1 [x (1) x(1) T 1 x ] T L = x M 0 ] 0. Two main approaches: 1 SDP relaxations, 2 SDP-induced valid inequalities. M = [νν T ] 0 α T Mα = [(α T ν) 2 ] L 0, α R n (orr n+1 ), α = 1. Dalkiran, Sherali (WSU, VT) RLT-based Optimization Software 15 / 25

38 ν-sdp cuts Model enhancements Semidefinite programming (SDP)-induced cuts (ν-sdp cuts) [xx T ] is symmetric and positive semidefinite M 0 = [xx T ] L 0. A stronger [ implication ] in this same vein is: [ 1 x (1) =, and defining the matrix M x 1 [x (1) x(1) T 1 x ] T L = x M 0 ] 0. Two main approaches: 1 SDP relaxations, 2 SDP-induced valid inequalities. M = [νν T ] 0 α T Mα = [(α T ν) 2 ] L 0, α R n (orr n+1 ), α = 1. 1 Let ( x, X) be a solution to the RLT relaxation. 2 Check if M 0, where M evaluates M at the solution ( x, X). Dalkiran, Sherali (WSU, VT) RLT-based Optimization Software 15 / 25

39 ν-sdp cuts Model enhancements Semidefinite programming (SDP)-induced cuts (ν-sdp cuts) [xx T ] is symmetric and positive semidefinite M 0 = [xx T ] L 0. A stronger [ implication ] in this same vein is: [ 1 x (1) =, and defining the matrix M x 1 [x (1) x(1) T 1 x ] T L = x M 0 ] 0. Two main approaches: 1 SDP relaxations, 2 SDP-induced valid inequalities. M = [νν T ] 0 α T Mα = [(α T ν) 2 ] L 0, α R n (orr n+1 ), α = 1. 1 Let ( x, X) be a solution to the RLT relaxation. 2 Check if M 0, where M evaluates M at the solution ( x, X). If not, we have an ᾱ R n+1 such that ᾱ T Mᾱ < 0. Append the SDP cut ᾱ T Mᾱ = [(ᾱ T ν) 2 ] L 0, and go to Step 1. Dalkiran, Sherali (WSU, VT) RLT-based Optimization Software 15 / 25

40 ν-vectors Model enhancements Semidefinite programming (SDP)-induced cuts (ν-sdp cuts) [ ] T ν (1) 1, {xj, j N}, {all quadratic monomials using x = j, j N},..., R ( n+ ). {all monomials of order using x j, j N } Dalkiran, Sherali (WSU, VT) RLT-based Optimization Software 16 / 25

41 ν-vectors Model enhancements Semidefinite programming (SDP)-induced cuts (ν-sdp cuts) [ ] T ν (1) 1, {xj, j N}, {all quadratic monomials using x = j, j N},..., R ( n+ ). {all monomials of order using x j, j N } where [ ν (2) 1, {xj, j N = J }, {all quadratic monomials using x j, j N J },..., {all monomials of order using x j, j N J } { J arg max J N r {0, 1,..., R} : J rt = J for some t Tr [ α rt XJrt ] } x j. j J rt ] T Dalkiran, Sherali (WSU, VT) RLT-based Optimization Software 16 / 25

42 Model enhancements Semidefinite programming (SDP)-induced cuts (ν-sdp cuts) ν-vectors [ ] T ν (1) 1, {xj, j N}, {all quadratic monomials using x = j, j N},..., R ( n+ ). {all monomials of order using x j, j N } [ ν (2) 1, {xj, j N = J }, {all quadratic monomials using x j, j N J },..., {all monomials of order using x j, j N J } ] T where { J arg max J N r {0, 1,..., R} : J rt = J for some t Tr [ α rt XJrt ] } x j. j J rt For a polynomial constraint φ r (x) β r of order δ r, define r δ 2 δr. If r 1, let 2 ν (3) = [1, all monomials of order r using x j, j N ] T. Then, we impose the following: { [φ r (x) β r ]ν (3) (ν (3) ) T } L 0. Dalkiran, Sherali (WSU, VT) RLT-based Optimization Software 16 / 25

43 Model enhancements Semidefinite programming (SDP)-induced cuts (ν-sdp cuts) ν-sdp cut inheritance parent ν-sdp inherit parent ν-sdp self Copy all Filter inactive cuts left child ν-sdp self left child ν-sdp inherit right child ν-sdp self right child ν-sdp inherit Dalkiran, Sherali (WSU, VT) RLT-based Optimization Software 17 / 25

44 Cut generation vs. branching Model enhancements Semidefinite programming (SDP)-induced cuts (ν-sdp cuts) i i+1 GAP branching = GAP left GAP parent max (1, UB ) i+2 GAP branching i+1 i+2 i+3 GAP branching i+4 GAP branching SDP cuts? i+3 i+4 Expected ( GAP i+3 SDP ) > GAP i+3 branching 2 Dalkiran, Sherali (WSU, VT) RLT-based Optimization Software 18 / 25

45 Model enhancements Semidefinite programming (SDP)-induced cuts (ν-sdp cuts) Table : Performances of SDP cut generation routines. Average CPU time (in seconds) Number of unsolved problems Degree No SDP Routine 1 Routine 2 Routine 3 Routine 4 No SDP Routine 1 Routine 2 Routine 3 Routine Routine 1: Generate SDP cuts and re-optimize the relaxation if they perform well. Else, generate and store SDP cuts for inheritance, if they perform well. Else, don t generate. Routine 2: Generate and store SDP cuts for inheritance, if they perform well. Else, don t generate. Routine 3: Generate SDP cuts and re-optimize the relaxation. Routine 4: Generate and store SDP cuts for inheritance. Dalkiran, Sherali (WSU, VT) RLT-based Optimization Software 19 / 25

46 CPU Time (in seconds) CPU Time (in seconds) Computational Results Problems without equality constraints Figure : Performance of the RLT algorithms for solving polynomial problems without equality constraints (in CPU seconds). RLT J-set Degree-two Problem Density Degree-four Degree-six Problem Density 0.0 Problem Density Dalkiran, Sherali (WSU, VT) RLT-based Optimization Software 20 / 25

47 CPU Time (in seconds) CPU Time (in seconds) Computational Results Problems with equality constraints Figure : Performance of the RLT algorithms for solving quadratic and cubic problems with equality constraints (in CPU seconds) Degree-two RLT-E J-Hybrid Proposed Degree-three RLT-E J-Hybrid Proposed 0.1 Dalkiran, Sherali (WSU, VT) RLT-based Optimization Software 21 / 25

48 CPU Time (in seconds) CPU Time (in seconds) Computational Results Problems with equality constraints Figure : Performance of the RLT Hybrid algorithms for solving degree-four, -five, -six, and -seven problems with equality constraints (in CPU seconds) Degree-four RLT-Hybrid J-Hybrid Proposed Degree-six RLT-Hybrid J-Hybrid Proposed Dalkiran, Sherali (WSU, VT) RLT-based Optimization Software 22 / 25

49 CPU Time (in seconds) CPU Time (in seconds) Computational Results Problems with equality constraints RLT-POS vs. BARON RLT-POS BARON Degree-two RLT-POS Degree-four 1000 RLT-POS Degree-six 100 BARON 100 BARON Dalkiran, Sherali (WSU, VT) RLT-based Optimization Software 23 / 25

50 Conclusions and future research directions Coordination between constraint filtering and reduced basis techniques. SDP cut generation routine for sparse problems. The J-Hybrid algorithm. RLT-based open-source optimization software. Dalkiran, Sherali (WSU, VT) RLT-based Optimization Software 24 / 25

51 Conclusions and future research directions Coordination between constraint filtering and reduced basis techniques. SDP cut generation routine for sparse problems. The J-Hybrid algorithm. RLT-based open-source optimization software. Nonlinear equality constraints. Tighten the relaxation in the reduced subspace. Stability of J-set of relaxations: Barrier and dual optimizer of CPLEX. Factorable programming problems and nonlinear integer programming problems. Dalkiran, Sherali (WSU, VT) RLT-based Optimization Software 24 / 25

52 Conclusions and future research directions Thank you! Dalkiran, Sherali (WSU, VT) RLT-based Optimization Software 25 / 25